Snakes, Swan’s Necks, and Puppy Dog Tails
It has been noted in passing that seven cervical vertebrae in
mammalian necks seems to be a threshold of sorts for movements
(refs).
If you add an additional link, the movements take on a
looser quality that does not
seem quite ‘neck-like” and
stiffening one linkage, as might happen with a fusion or
arthritic restrictions, leads to a substantial reduction in
movement.
One might speculate that the number of vertebrae and
the amount of movement in the intervertebral linkages might be
in some way set by evolution to achieve the needs of a mammalian
neck (ref).
With a very small number of exceptions (manatees,
and two types of sloth), all mammals have 7 cervical vertebrae.
A snake’s skeleton has a series of nearly identical
elements composed or a vertebra and two attached ribs.
The concatenation of many small movements between
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successive vertebrae allows the snake to take a wide
variety of sinuous shapes.
That leads to curiosity about the implications of chains of
many similar linkages, such as occur in a cat’s tail, a snake’s
body, or a swan’s neck.
cervical vertebrae.
A swan’s neck may have up to 25
Some of the aquatic reptiles of the
Mesozoic, plesiosaurs, had up to 40 cervical vertebrae.
cat’s tail may have 21-23 caudal vertebrae.
The
Snakes have largely
identical series of vertebral segments, with attached ribs.
All
of these anatomical structures have a great deal of mobility and
a similar manner of moving.
In this chapter, we will consider the dynamics of a mechanical
system that has a high degree of repetition of concatenated
identical elements.
While we start with the idea of a snake or
a swan’s neck, we will be concerned with an abstract entity that
we will call an artificial snake.
The results may not be
directly applicable to actual snakes or swans necks or puppy dog
tails, because a close examination will reveal that the elements
in those structures are not in fact identical.
come closest.
Snake spines
Bird necks are most differentiated.
Artificial Snakes
We will talk about joints and muscles and how muscles move
bones at joints.
While we start with the concepts of actual
joints and muscles, we will again recast those entities into
abstractions and consider the formal properties of groups of
muscles and joints working in concert.
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An artificial snake is created by linking a series of
identical elements that can move about a central pivot
point. This mechanism can be described abstractly by
set of linked framed vectors.
We will start with a very simple toy.
A toy snake is
constructed by concatenating a number of identical elements
[ E1 , E 2 , E 3 ,…, E n ].
Each has a pivot point, which is taken to be
the location [P] of the element.
In the illustrated instance
all the pivot points have an axis of rotation [p] that is
directed perpendicular to the reference plane [p = r] and the
pivot points are linked to each other in directions that extend
parallel to the reference plane.
The linkage [L] is an
extension of the element to the next pivot point and it defines
a direction relative to the element [s].
The direction of the
link and the direction of the pivot axis define an orientation
frame [O].
We will assume that the mutually perpendicular
direction that your thumb points when the fingers of your right
hand curl from the pivot axis to the linkage direction will the
final axis of the orientation frame [t].
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The frame vectors of a single element
In addition, we will have two armatures that extend away from
the pivot point in the plane of the pivot axis, that is,
parallel with the reference plane.
The two armatures [ A1, A2 ]
will extend symmetrically to either side of the linkage so as to
form a right angle between them.
and a 2 .
Their directions will be a 1
Their length will be ‘  ’.
Therefore, each element of
the snake may be written as a framed vector.


E n = Pn,L n, A1n, A2n, rn, sn, tn  ;
Ln   sn ,
A1n   a1n and A2n   a 2n .
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The abstract toy snake is expressed by a series of
linked elements composed of a pivot point at a
particular location and two extended vectors that arise
from the pivot point and extend symmetrically to either
side of the axis of the element. The element is
orientable. Each element can be described by a framed
vector.
The rotation between elements is given by the quaternion
R n  R n ,p  cosn  p sinn  cosn  r sinn .
We can write down the expression for the snake from these
expressions.


E n+1 = Pn + L n,L n+1, A1n+1, A2n+1, rn, R n+1  sn, R n+1  tn  ;
L n+1  R n+1   sn ,
A1n+1  R n+1   a1n and A2n+1  R n+1   a 2n .
We can place it in space by specifying the location and
orientation of the first element.


E 1 = P1,L1, A11, A21, r1, s1, t1 ;
L 1   s1 ,
A11   a11 and A21   a 21 ,
  
  
a11  R   , r   s and a11  R   , r   s .
 4 
 4 
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Since the axis of the pivot joint is fixed relative to the
element and it is aligned with the r axis, the calculation is
very straight-forward.
We can also use the full angle form of
quaternion multiplication to compute the new values of the frame
elements.
The pivot axis is allowed to take any direction
relative to the element. Since the pivot axis is no
longer perpendicular to the plane of the element, the r
vector of the frame may be defined as the direction of
the ratio of two of the extension vectors.
Having carried the analysis to this point in an essentially
two-dimensional situation, it is easy to write down the
description for an arbitrary snake in three dimensions.
The
main difference is that the pivot axis can take on any
direction, therefore, we need to use the half angle expression
for rotation.
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

E n+1 = Pn+1,L n+1, A1n+1, A2n+1, rn+1, s n+1, tn+1


;
-1
-1
-1
= Pn + L n ,L n+1, A1n+1, A2n+1, n+1  rn  n+1
, n+1  sn  n+1
, n+1  tn  n+1


n+1  cos  pn+1 sin ,


-1
L n+1  n+1   sn  n+1 ,
-1
-1
A1n+1  n+1   a1n  n+1
and A2n+1  n+1   a 2n  n+1
,
 

 

a1n+1  R   , rn+1   s and a 2n+1  R   , rn+1   s ,
 4

 4

a1n+1 =
A1n+1
A1n+1
,
A 
 L 
A 
r = UV  2n  = UV  2n  = UV 
.
 L 
 A1n 
 A1n 
More generally, create a real or hypothetical vertical or
lateral extension that is moved with the element.
The extension elements may be written as functions of the
orientation frame of reference.
Then, we need only keep track
or the locations and orientations.
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